Cohomology rings of the plactic monoid algebra via a Gröbner–Shirshov basis

Abstract

In this paper we calculate the cohomology ring Ext * kPln (k, k) and the Hochschild cohomology ring of the plactic monoid algebra kPln via the Anick resolution using a Gröbner -Shirshov basis. IntroductionThe plactic monoid was discovered by Knuth [15], who used an operation given by Schensted in his study of the longest increasing subsequence of a permutation. It was named and systematically studies by Lascoux and Schützenberger [22], who allowed any totally ordered alphabet in the definition. It is known that the elements of plactic monoid can be written in the canonical form, and in this form can be identified with some type of the Young tableaux. Because of its strong relations to Young tableaux, the plactic monoid has already become a classical tool in several areas of representation theory and algebraic combinatorics [20].Among the significant applications are: a proof of the Littlewood-Richardson rule, an algorithm which allows to decompose tensor product of representations of unitary groups, a combinatorial description of the Kostka -Foulkes polynomials, which arise as entries of character table of the finite linear group. The plactic monoid appeared also in: theory of modular representations of the symmetric group, quantum groups, via the representation theory of quantum enveloping algebras. It is worth mentioning that even though the combinatorics of the plactic monoid has been extensively studied, there are only a few preliminary results of the corresponding plactic algebra over a field [9].The plactic monoid was connected to the free 2-nilpotent Lie algebra (which is a subalgebra of the parastatistics algebra) in the work [24] by J-L. Loday and T. Popov. The connection is through the quantum deformation (in the sense of Drinfeld) of the parastatistics algebra. But the first work where the connection between the plactic monoid algebra and parastastics algebra (in two dimensions) was found is [30].In [5] there was given an independent proof of uniqueness of (Robinson -Shensted) Knuth normal forms of elements of (Knuth -Schützenberger) plactic monoid.Gröbner bases and Gröbner -Shirshov bases were invented independently by A.I. Shirshov for ideals of free (commutative, anti-commutative) non-associative algebras [17,28], free Lie algebras [27,28] by H. Hironaka [13] for ideals of the power series algebras (both formal and convergent), and by B. Buchberger [6] for ideals of the polynomial algebras.Cain et al [7] use the Schensted-Knuth normal form (the set of (semistandart) Young tableaux) to prove that the multiplication table of column words (strictly decreasing words with respect to some order on the letters) forms a finite Gröbner -Shirshov basis of the finitely generated plactic monoid. In [5] were given new explicit formulas for the multiplication tables of row (nondecreasing word) and column words and independent proofs that the resulting sets of relations are Gröbner -Shirshov bases in row and column generators respectively.The Anick resolution was obtained by David J. Anick in 1986 [1]. This is a r...

Finite convergent presentation of plactic monoid for type C

“…A coherent presentation of a monoid is a first step to a polygraphic resolution of the monoid, that is, a categorical cofibrant replacement of the monoid which can be used to compute its homological invariants [GM12]. In [Lop14], Lopatkin constructed Anick's resolution for the monoid P n (A) starting with a finite convergent presentation. Our finite convergent presentation of the monoid P n (C) should allow us to compute a polygraphic resolution of it which is a generalisation of Anick's resolution.…”

Knuth’s Coherent Presentations of Plactic Monoids of Type A

“…More generally, one may conjecture that the generating n-cells of the resolution have the form of the permutohedron of order n corresponding to a confluence diagram of (n − 1) overlapping reductions. This construction should generalise the construction of the Anick resolution for the monoid P n starting with the column presentation, given by Lopatkin in [30].…”

The Hochschild cohomology of the group $G^2_3$

“…In this section, we state main definitions and essential results related to the construction of the Anick chain complex via algebraic Morse theory following [1,2,4,6,7,8]. Let k be a field and let Λ be a unital associative k-algebra with an augmentation, i.e., a k-algebra homomorphism ε : Λ → k. Let A be a set of generators for Λ.…”

“…A series of groups G k n were introduced in [12,13,14,15] in the study of braids, links, and Coxeter groups. In this paper, we consider the group G 2 3 originated in the following (informal) context. Assume three points are moving without collisions on a disk in the plane at time t ∈ [0, 1].…”

“…In the present work, we apply the Morse matching theory to find the Anick resolution and calculate the groups of Hochschild n-cohomologies of the group G 2 3 for all n ≥ 1 with coefficients in all 1-dimensional kG 2 3bimodules. It is easy to see that all such modules are of the form W = ǫ 1 W ǫ 2 , where…”